It’s an old and oft-asked question: is mathematics ultimately something that is invented, or something that is discovered? Or, put another way: is mathematics something arbitrarily set up by us humans, or something inevitable and fundamental and in a sense “preexisting”, that we merely get to explore? In the past it’s seemed as if these were two fundamentally incompatible possibilities. But the framework we’ve built here in a sense blends them both into a rather unexpected synthesis.
The starting point is the idea that mathematics—like physics—is rooted in the ruliad, which is a representation of formal necessity. Actual mathematics as we “experience” it is—like physics—based on the particular sampling we make of the ruliad. But then the crucial point is that very basic characteristics of us as “observers” are sufficient to constrain that experience to be our general mathematics—or our physics.
At some level we can say that “mathematics is always there”—because every aspect of it is ultimately encoded in the ruliad. But in another sense we can say that the mathematics we have is all “up to us”—because it’s based on how we sample the ruliad. But the point is that that sampling is not somehow “arbitrary”: if we’re talking about mathematics for us humans then it’s us ultimately doing the sampling, and the sampling is inevitably constrained by general features of our nature.
A major discovery from our Physics Project is that it doesn’t take much in the way of constraints on the observer to deeply constrain the laws of physics they will perceive. And similarly we posit here that for “observers like us” there will inevitably be general (“physicalized”) laws of mathematics, that make mathematics inevitably have the general kinds of characteristics we perceive it to have (such as the possibility of doing mathematics at a high level, without always having to drop down to an “atomic” level).
Particularly over the past century there’s been the idea that mathematics can be specified in terms of axiom systems, and that these axiom systems can somehow be “invented at will”. But our framework does two things. First, it says that “far below” axiom systems is the raw ruliad, which in a sense represents all possible axiom systems. And second, it says that whatever axiom systems we perceive to be “operating” will be ones that we as observers can pick out from the underlying structure of the ruliad.
At a formal level we can “invent” an arbitrary axiom system (and it’ll be somewhere in the ruliad), but only certain axiom systems will be ones that describe what we as “mathematical observers” can perceive. In a physics setting we might construct some formal physical theory that talks about detailed patterns in the atoms of space (or molecules in a gas), but the kind of “coarse-grained” observations that we can make won’t capture these. Put another way, observers like us can perceive certain kinds of things, and can describe things in terms of these perceptions. But with the wrong kind of theory—or “axioms”—these descriptions won’t be sufficient—and only an observer who’s “shredded” down to a more “atomic” level will be able to track what’s going on.
There’s lots of different possible math—and physics—in the ruliad. But observers like us can only “access” a certain type. Some putative alien not like us might access a different type—and might end up with both a different math and a different physics. Deep underneath they—like us—would be talking about the ruliad. But they’d be taking different samples of it, and describing different aspects of it.
For much of the history of mathematics there was a close alignment between the mathematics that was done and what we perceive in the world. For example, Euclidean geometry—with its whole axiomatic structure—was originally conceived just as an idealization of geometrical things that we observe about the world. But by the late 1800s the idea had emerged that one could create “disembodied” axiomatic systems with no particular grounding in our experience in the world.
And, yes, there are many possible disembodied axiom systems that one can set up. And in doing ruliology and generally exploring the computational universe it’s interesting to investigate what they do. But the point is that this is something quite different from mathematics as mathematics is normally conceived. Because in a sense mathematics—like physics—is a “more human” activity that’s based on what “observers like us” make of the raw formal structure that is ultimately embodied in the ruliad.
When it comes to physics there are, it seems, two crucial features of “observers like us”. First, that we’re computationally bounded. And second, that we have the perception that we’re persistent—and have a definite and continuous thread of experience. At the level of atoms of space, we’re in a sense constantly being “remade”. But we nevertheless perceive it as always being the “same us”.
This single seemingly simple assumption has far-reaching consequences. For example, it leads us to experience a single thread of time. And from the notion that we maintain a continuity of experience from every successive moment to the next we are inexorably led to the idea of a perceived continuum—not only in time, but also for motion and in space. And when combined with intrinsic features of the ruliad and of multicomputation in general, what comes out in the end is a surprisingly precise description of how we’ll perceive our universe to operate—that seems to correspond exactly with known core laws of physics.
What does that kind of thinking tell us about mathematics? The basic point is that—since in the end both relate to humans—there is necessarily a close correspondence between physical and mathematical observers. Both are computationally bounded. And the assumption of persistence in time for physical observers becomes for mathematical observers the concept of maintaining coherence as more statements are accumulated. And when combined with intrinsic features of the ruliad and multicomputation this then turns out to imply the kind of physicalized laws of mathematics that we’ve discussed.
In a formal axiomatic view of mathematics one just imagines that one invents axioms and sees their consequences. But what we’re describing here is a view of mathematics that is ultimately just about the ways that we as mathematical observers sample and experience the ruliad. And if we use axiom systems it has to be as a kind of “intermediate language” that helps us make a slightly higher-level description of some corner of the raw ruliad. But actual “human-level” mathematics—like human-level physics—operates at a higher level.
Our everyday experience of the physical world gives us the impression that we have a kind of “direct access” to many foundational features of physics, like the existence of space and the phenomenon of motion. But our Physics Project implies that these are not concepts that are in any sense “already there”; they are just things that emerge from the raw ruliad when you “parse” it in the kinds of ways observers like us do.
In mathematics it’s less obvious (at least to all but perhaps experienced pure mathematicians) that there’s “direct access” to anything. But in our view of mathematics here, it’s ultimately just like physics—and ultimately also rooted in the ruliad, but sampled not by physical observers but by mathematical ones.
So from this point of view there’s just as much that’s “real” underneath mathematics as there is underneath physics. The mathematics is sampled slightly differently (though very similarly)—but we should not in any sense consider it “fundamentally more abstract”.
When we think of ourselves as entities within the ruliad, we can build up what we might consider a “fully abstract” description of how we get our “experience” of physics. And we can basically do the same thing for mathematics. So if we take the commonsense point of view that physics fundamentally exists “for real”, we’re forced into the same point of view for mathematics. In other words, if we say that the physical universe exists, so must we also say that in some fundamental sense, mathematics also exists.
It’s not something we as humans “just make”, but it is something that is made through our particular way of observing the ruliad, that is ultimately defined by our particular characteristics as observers, with our particular core assumptions about the world, our particular kinds of sensory experience, and so on.
So what can we say in the end about whether mathematics is “invented” or “discovered”? It is neither. Its underpinnings are the ruliad, whose structure is a matter of formal necessity. But its perceived form for us is determined by our intrinsic characteristics as observers. We neither get to “arbitrarily invent” what’s underneath, nor do we get to “arbitrarily discover” what’s already there. The mathematics we see is the result of a combination of formal necessity in the underlying ruliad, and the particular forms of perception that we—as entities like us—have. Putative aliens could have quite different mathematics, not because the underlying ruliad is any different for them, but because their forms of perception might be different. And it’s the same with physics: even though they “live in the same physical universe” their perception of the laws of physics could be quite different.